Magnificent Mistakes in Mathematics

Magnificent Mistakes in Mathematics

by Alfred S. Posamentier, Ingmar Lehmann

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Two veteran math educators demonstrate how some "magnificent mistakes" had profound consequences for our understanding of mathematics' key concepts.

In the nineteenth century, English mathematician William Shanks spent fifteen years calculating the value of pi, setting a record for the number of decimal places. Later, his calculation was reproduced


Two veteran math educators demonstrate how some "magnificent mistakes" had profound consequences for our understanding of mathematics' key concepts.

In the nineteenth century, English mathematician William Shanks spent fifteen years calculating the value of pi, setting a record for the number of decimal places. Later, his calculation was reproduced using large wooden numerals to decorate the cupola of a hall in the Palais de la Découverte in Paris. However, in 1946, with the aid of a mechanical desk calculator that ran for seventy hours, it was discovered that there was a mistake in the 528th decimal place. Today, supercomputers have determined the value of pi to trillions of decimal places.

This is just one of the amusing and intriguing stories about mistakes in mathematics in this layperson's guide to mathematical principles. In another example, the authors show that when we "prove" that every triangle is isosceles, we are violating a concept not even known to Euclid - that of "betweenness." And if we disregard the time-honored Pythagorean theorem, this is a misuse of the concept of infinity. Even using correct procedures can sometimes lead to absurd - but enlightening - results.

Requiring no more than high-school-level math competency, this playful excursion through the nuances of math will give you a better grasp of this fundamental, all-important science.

From the Hardcover edition.

Editorial Reviews

Library Journal
Posamentier (mathematics education, Mercy Coll.) and Lehmann (formerly, mathematics, Humboldt Univ., Berlin), coauthors of several math books, present a compendium of mathematical errors, arranged by topic into five chapters. Only the first, on errors made by eminent mathematicians, has much narrative. Here, we see number-theoretic conjectures that were based upon a few examples and a lot of wishful thinking. In Chapter Two, the authors address errors in arithmetic based mostly upon misunderstandings of notations. Chapter Three deals with algebraic errors primarily caused by inadvertent division by zero or the introduction of extraneous roots. In Chapter Four we have errors in geometric proofs, mostly arising from deliberately misleading diagrams. The final chapter deals with probability, showing mistakes made by confusing the concepts of conditionality, independence, and mutual exclusivity of events. Many of the errors are clever and instructive, some are repetitive and uninteresting; only a few should be designated as "magnificent." VERDICT This volume could be useful to a teacher of mathematics as a source of examples that can hammer home important concepts. Beyond that, it will have a limited readership.—Harold D. Shane, Mathematics, Emeritus, Baruch Coll., CUNY
From the Publisher
“[F]or me it is the mathematical gaffes the rest of us are more likely to fall for that make Posamentier and Lehmann's book interesting. My favourites are the proofs that involve dividing by zero, leading to results such as 1=2.”

 -New Scientist 

“This volume could be useful to a teacher of mathematics as a source of examples that can hammer home important concepts.”

-Library Journal

“Don’t be scared off by the ‘mathiness’… Magnificent Mistakes in Mathematics is an intriguing read. I’ll bet that even old hands at mathematics will find something new here… a great opportunity as the school year begins, giving us a chance once again to ‘go figure!’” 

-Astro Guyz

“A fairly quick and entertaining read for typical math buffs… I think this somewhat unique approach makes the volume a worthwhile, entertaining addition to one's math bookshelf, and it may be particularly useful to secondary school teachers, providing a lot of grist for instructive, thoughtful examples in the classroom.”


“Advanced math skills are not required to enjoy untangling errors that illustrate concepts in arithmetic, algebra, and more.”
-Science News 

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By Alfred S. Posamentier, Ingmar Lehmann

Prometheus Books

Copyright © 2013 Alfred S. Posamentier and Ingmar Lehmann
All rights reserved.
ISBN: 978-1-61614-748-8

<h2>CHAPTER 1</h2> <p><b>NOTEWORTHY MISTAKES BY FAMOUS MATHEMATICIANS</b></p> <br> <p>There are numerous conjectures by famous and less-famous mathematicians that have been published over the years. Some of these conjectures have been subsequently supported by proof, while some have been dismissed as mistaken, and others are still seeking verification or dismissal. Yet in all cases the attempts to grapple with these conjectures have moved our understanding of mathematics to the next level. Our journey through some of these conjectures will be to see what it took to verify them and what might have been found to dismiss them.</p> <p>With a broad overview we can see that some of the greatest thinkers—mathematicians and scientists—have made mistaken conjectures, many of which have led to new discoveries and a broadening of their respective fields. For example, Aristotle (384–322 BCE), one of the most influential and well-known thinkers of all time, has made some mistaken statements. Although one might consider him one of the founders of science, and what he wrote has influenced generations, his errors have opened new fields of thought and study. Here a few of his mistaken beliefs:</p> <p>• The world consists of five elements: fire, water, air, earth, and ether. The first four are nature on earth, and the fifth fills the heavens.</p> <p>• Heavier weights fall to earth faster than lighter ones—yet because of Galileo Galilei (1564–1642) we know better.</p> <p>• Flies have four legs.</p> <p>• Women have fewer teeth than men.</p> <br> <p>Aristotle also believed that the Earth was the center of the universe, and that the other observable bodies—such as the moon, the sun, and the planets—revolve around it. His words were quite influential in his time and lasted quite some time beyond. He influenced some of the other great thinkers such as Hipparchus of Nicaea (ca. 190–120 BCE) and Claudius Ptolemy (ca. 100–before 180). Not until Nicolaus Copernicus (1473–1543), Tycho Brahe (1546–1601), and Johannes Kepler (1571–1630) proved that the universe was solar centric did these earlier beliefs lose favor. Originally Kepler believed that the planets traveled on the surface of a sphere—with a circular path—so as to be boxed in by the platonic solids. This was nicely supportive of Pythagoras's beliefs even though they were false! Later on, he corrected his erroneous conjecture by stating that the planets traveled on an elliptical path and described through his three famous laws the nature of the planet's travel along the elliptical path—perhaps one of the greatest achievements in astronomy and mathematics, and one that has solidified his fame forever.</p> <p>Despite his enlightened thinking, Kepler still had a weakness for astrology. Similarly, Newton had a fascination with alchemy and a deep belief in religion and mystiques of numerology. This latter interest motivated Newton to produce thousands of pages of numerological calculations, which led him to predict that the world will come to an end in the year 2060. Was this a mistake?</p> <p>While speaking of the earth, consider the most important mistake in the history of America. The Italian explorer Christopher Columbus (1451–1506), through his calculations, was convinced that the western route to India was considerably shorter than that in the easterly direction. This belief was based on the mistaken measurement by the astronomer Claudius Ptolemy, who calculated the circumference of the Earth to be 28,000 km. With this estimation for the earth's circumference, Columbus's conjecture would have been correct. However, the experts in the Spanish court assessed the earth's circumference at 39,000 km, which was arrived at by the Italian mathematician and cartographer Paolo Toscanelli (1397–1482). This calculation was incredibly accurate for the times, as it is quite close to the actual circumference of 40,075 km. Thus, Columbus's calculation was destroyed in the Spanish court. It was also common knowledge at the time that the earth was spherical, and thus Columbus was not trying to prove that it wasn't flat.</p> <p>Some mistakes are—by today's knowledge—downright silly. The British physicist William Thompson (1824–1907)—perhaps better known as Lord Kelvin, after whom the Kelvin temperature scale is named—believed that there would never be an airplane that would be heavier than air!</p> <p>The famous Austrian psychologist Sigmund Freud (1856–1939) was always fascinated with the mystique of numbers. Despite his brilliance, he was enchanted with the conjecture offered by the German biologist Wilhelm Fliess (1858–1928) that any number expressible by thee combination of multiples 23 and 28 (or put another way that could be expressed as 23&llt;i>x</i> + 28<i>y</i>), had some special significance in a person's life cycle. He then cccclaimed that many people die at age 51, and found that 23 · 1 + 28 · 1 = 51. Yet, 23 · 3 + 28 · (-2) = 13—not usually considered a desirable number! The fact that almost all numbers can be expressed by 23<i>x</i> + 28<i>y</i> did not occur to Freud. Another rather silly mistake!</p> <p>Let us now consider some mathematical mistakes made by some magnificent mathematicians.</p> <br> <p><b>PYTHAGORAS'S MISTAKE</b></p> <p>Pythagoras of Samos (ca. 570 BCE–510 BCE) is most famous for the relationship bearing his name concerning the sides of a right triangle. Unfortunately, we do not have any of his writings, but there is much attributed to him, nevertheless. Today, we have strong evidence that even his famous theorem was known to the Babylonians and the Egyptians in the special case of a triangle with side lengths 3, 4, and 5 hundreds of years earlier.</p> <p>A society centered on Pythagoras evolved that was fascinated with numbers, and it was felt that everything could be explained with numbers—namely, the natural numbers: 1, 2, 3, 4, 5,.... The belief was that principles of mathematics were the principles that explained the world. Harmony and nature were to have been explained through these numbers. For example, in music, intervals can be determined by number relationships. In all these instances, he was able to explain the relationships through the natural numbers and their ratios. Wanting to extend this to geometric forms, Pythagoras erred.</p> <p>One of his society members, Hippasus of Metapontum (ca. fifth century BCE) dispelled Pythagoras's belief that the dimensions of a regular pentagram could also be measured with natural numbers, by discovering that there must be other numbers (later developed as the irrational numbers) to explain the relationships of the various segment lengths of a pentagram (see figures 1.1 and 1.2). This was shocking to Pythagoras, perhaps because this geometric figure was the symbol of the Pythagoreans.</p> <p>The relationship between the segments of a pentagram has some very interesting properties: <i>d/a = a/e = e/f</i> (see figure 1.3).</p> <p>This relationship relates to the famous golden ratio,</p> <p>φ = [√5 + 1]/2 ≈ = 1,618033988749894848204586834365638117720.</p> <p>As a postscript to correcting this mistake of Pythagoras, Hippasus is supposed to have drowned as punishment for this "sacrilege"—extending beyond the natural numbers, which were supposed to define everything!</p> <br> <p><b>MISTAKES IN THE ORIGINAL LOGARITHMS TABLES</b></p> <p>In 1614, the Scottish mathematician John Napier (1550–1617) published a book on logarithms, which was recognized as particularly significant by the English mathematician Henry Briggs (1561–1630), who, in 1624, consequently published a table to logarithms to fourteen decimal places for the base 10 numbers, titled <i>Arithmetica logarithmica</i>. Curiously, Briggs's table of logarithms contained the numbers from 1 to 100,000, but omitted the numbers from 20,000 to 90,000. This void was filled by the Dutch mathematician Adrian Vlacq (1600–1667), who in addition to being a good mathematician was also a clever book merchant. Furthermore, he was particularly important in the popularization of the use of logarithms. His complete table of logarithms first appeared in 1628, as a second edition of Briggs's <i>Arithmetica logarithmica</i>, and provided a complete listing of the logarithms of numbers from 1 to 100,000, but only up to ten decimal places.</p> <p>In 1794, an improved printing of <i>Vlacq's Table</i> was published in Leipzig, Germany, by Baron Jurij Bartolomej Vega (Latin: <i>Georgius Bartholomaei Vecha</i>; German: <i>Georg Freiherr von Vega</i>; 1754–1802), which became the model for all future number tables. In the preface for this publication, the author stated that the first person to find any mistakes in his tables, which could lead to a faulty calculation, would be paid in ducats. Needless to say, it would have been quite astonishing if this table were fault-free. In the course of time, about three hundred mistakes were discovered—and not merely in the final decimal place!</p> <br> <p><b>FERMAT'S LAST THEOREM</b></p> <p>One of the most famous conjectures is the longtime famous Fermat's last theorem, which states that <i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i> has no (non-zero) integer solutions for <i>n</i> > 2. Of course, we know that for <i>n</i> = 1 this is trivial, and for <i>n</i> = 2 we have the Pythagorean theorem. This conjecture by the famous French mathematician Pierre de Fermat (1607/08–1665) was made in 1637 and was finally proved by Andrew Wiles (and Richard Taylor) in 1995. So for 358 years, Fermat's comments in the margin of one of his books, Diophantus's <i>Arithmetica</i>, stood unproven, where he wrote that</p> <p>it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second power into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.</p> <br> <p>Today, some assume that Fermat had a proof for this special case where <i>n</i> = 4 and then figured it could be generalized for all values of <i>n</i>. Some number theorists today doubt that Fermat ever did have a proof for all values of <i>n</i> > 2.</p> <p>Along the long path from 1637 to 1995 there has been a series of many mistaken attempts where people thought they found the proof that Fermat considered and others presented proofs of their own invention that also were discovered to have a mistake. The attraction of this 358-year-old puzzle lies in the simplicity of its statement, so that laymen and famous mathematicians such as Leonhard Euler (1707–1783), Ernst Eduard Kummer (1810–1893), Carl Friedrich Gauss (1777–1855), and Augustin Louis Cauchy (1789–1857) attempted to solve this problem—each time with mistakes.</p> <p>In 1770, Euler showed that the equation <i>x</i><sup>3</sup> + <i>y</i><sup>3</sup> = <i>z</i><sup>3</sup> has no solution among the natural numbers. Yet this proof was not complete and required some help in 1830 by the French mathematician Adrien-Marie Legendre (1752–1833). Gauss also provided a correct solution for the case where <i>n</i> = 3. We should note that in 1738 Euler successfully dealt with the case where <i>n</i> = 4. However, later on it was discovered that the case where <i>n</i> = 4 was already discovered in 1676 by Bernard Frenicle de Bessy (ca.1605–1675). After Euler's death there were many futile attempts to solve Fermat's conjecture, notably one by the French mathematician Sophie Germain (1776–1803), who, as a woman living at the time, was forced to publish under the pseudonym Monsieur Le Blanc and set the stage for the proof that for n = 5, Fermat's conjecture held true.</p> <p>Yet, despite the mistaken attempts, these did help further develop algebraic number theory. In 1828, Peter Lejeune Dirichlet (1805–1859) and Adrien-Marie Legendre were able to show that the equation <i>x</i><sup>5</sup> + <i>y</i><sup>5</sup> = <i>z</i><sup>5</sup> does not have an integer solution. In this process, Dirichlet made a mistake that was eventually corrected by Legendre. The mistake in Dirichlet's proof meant that it was not complete; but with Legendre's help, it was correctly completed.</p> <p>This process of proof attempts to support Fermat's conjecture continued, such that in 1839 Gabriel Lamé (1795–1870) proved that <i>x</i><sup>7</sup> + <i>y</i><sup>7</sup> = <i>z</i><sup>7</sup> does not have an integer solution, a result that was also shown independently by Victor A. Lebesgue (1791–1875). In 1841, Lamé mistakenly thought that he had proved the general case—that which Fermat stated, since the case for <i>n</i> = 14 was put to rest in 1832 by Dirichlet. Finally, in 1847 some top mathematicians such as Cauchy and Lamé were (mistakenly) convinced that the general case of <i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i> was finally proved and were ready to present it to the French Academy of Science. However, Ernst Eduard Kummer destroyed their hopes for fame because he found a mistake in their work.</p> <p>The drama continued, where in 1905 a Goldmark prize worth 100,000 marks was offered by Paul Friedrich Wolfskehl (1856–1906) for a correct solution of this long-standing problem. This certainly enticed a multitude of mathematicians. On March 9 and 10, 1988, an article in the <i>Washington Post</i> and the <i>New York Times</i> reported that the thirty-year-old Japanese mathematician Yoichi Miyaoka finally proved Fermat's conjecture. However, in short order this highly vaunted attempt also was discovered to have mistakes.</p> <p>In 1993, during a workshop at the Isaac Newton Institute at Cambridge University, the British mathematician Andrew Wiles (1953–) over several days presented a proof that appeared to have been finally a correct proof of Fermat's last theorem. However, once again, soon afterward, Nicholas Katz (1943–) found a mistake in Wiles's proof. Wiles, along with his doctoral student Richard Taylor, then spent the next year feverishly trying to eradicate this error. On September 19, 1994, the mistake was corrected and, finally, Andrew Wiles had conquered this famous 358-year-old mathematical challenge. In June 1997, over a thousand scientists gathered at the University of Göttingen (Germany) as the Wolfskehl Prize (worth about $25,000) was awarded to Andrew Wiles, and with this he ended his tenyear journey to the solution of the famous Fermat conjecture. Despite the many mistakes made on the path to success, many interesting by-products in mathematics were discovered, which shows that sometimes mistakes in mathematics can be considered magnificent because they provide some valuable new mathematical insights.</p> <br> <p><b>GALILEO GALILEI'S BIG MISTAKE!</b></p> <p>The famous mathematician, physicist, and astronomer Galileo Galilei (1564–1642) concerned himself with uniformly accelerated motion for over forty years. His experimental innovation consisted of using the oblique plane, which he was able to use to study the laws of motion and was able to test them from a quantitative standpoint. In 1638, while searching for the fastest connection that a weight would travel between two points under the influence of gravity and without friction, he encountered a mistake. He noticed the time from <i>A</i> to <i>B</i> along a certain polygonal path was faster than that along line <i>AB</i> (see figure 1.4).</p> <p>He calculated this speed to make the so-called Galileo pendulum, with which he was able to establish the characteristics of a swinging ball. He noticed that the time of the ball's travel was inversely proportional to the number of vertices of the polygonal path. The more vertices the path has, the less travel time is required. Recognizing that by constantly increasing the number of vertices, the polygon surface will approach the arc of a circle, he therefore conjectured that the arc of a circle must be the fastest curve for the ball to travel rather than a straight line.</p> <p>What he didn't consider was that the polygon sides do not necessarily have the same length, and, therefore, do not approach the curve of the circle.</p> <p>Supposing we consider the point <i>P</i> as shown in figure 1.4 to be below point <i>B</i>, which is something that did not even occur to Galileo, then possibly the speed through the steepness of <i>AP</i> will compensate for this segment <i>PB</i>. In 1696, Johann Bernoulli (1667–1748) also investigated this problem. He, along with Mersenne, in 1644; Huygens, in 1657; and Lord Brouncker, in 1662, found the solution in <i>Brachistochrone</i>. When we look back today, we see the calculus of variations embedded in this situation. Bernoulli also considered the polygonal path as the ideal path, for he felt segments should be longer near the point <i>A</i> than the point <i>B</i>. This alteration prevented him from drawing the same conclusion as Galileo, since his path would not approach the curve of a circle. So we can see that while the solution that Galileo proposed was a good approximation, it did not give us the exact curve that we would require. Galileo's "false Brachistochrone" is, therefore, an arc of a circle!

Excerpted from MAGNIFICENT MISTAKES IN MATHEMATICS by Alfred S. Posamentier, Ingmar Lehmann. Copyright © 2013 Alfred S. Posamentier and Ingmar Lehmann. Excerpted by permission of Prometheus Books.
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Meet the Author

Alfred S. Posamentier is dean of the School of Education and professor of mathematics education at Mercy College in Dobbs Ferry, New York. Previously, he had the same positions at the City College of the City University of New York for forty years.  He has published over fifty-five books in the area of mathematics and mathematics education, including The Fabulous Fibonacci Numbers (with Ingmar Lehmann). 
Ingmar Lehmann is retired from the mathematics faculty at Humboldt University in Berlin. For many years he led the Berlin Mathematics Student Society for gifted secondary-school students, with which he is still closely engaged today.  He is the coauthor with Alfred S. Posamentier of The Secrets of TrianglesThe Glorious Golden Ratio, and three other books.

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